Let f(x) be a polynomial of degree four having extreme vlaues at x=1 and x =2 .If `lim_(x rarr 0) (1+(f(x))/(x^2))=3` then f(2) is equal to

Options

A. 0
B. 4
C. `-8`
D. `-4`

Answer

A

Explanation

If is given that f(x) is a fourth degree polynomial such that
`underset(xrarr0)lim(1+(f(x))/(x^))=3`
`rArr underset(xrarr0)lim(f(x))/(2)` is finite
`rArr f(x)` has a repeated root at x=0 ,
Let `f(x)=x^2(ax^2+bx+c)` Then
`underset(xrarr0)lim(1+(f(x)/(x^2))=3`
`rArr underset(x rarr 0)lim(1+ax^2+bx+c)=3`
`rArr c+1=3`
`rArr c=2`
If is given that f(x)has extreme values at x=1 and x=2
`therefore f(1)=0 and f(2)=0`
`rArr 4a+3b+2c=0 32a+12b+4c =0`
`[ because f(x)=4ax^2+3bx^2+2cx]`
`rArr 4a+3b+4=0 and 32 a+12b+8=0`
`rArr a=1//2,b=-2`
`therefore f(x)=x^2(1/2x^2-2x+2)`
`rArr f(2)=4(2-4+2)=0`